find the equation of the tangent line to the graph of f(x) = cos3xln(x^4) at (1,0)
Can someone please guide me through this problem?

Question

find the equation of the tangent line to the graph of f(x) = cos3xln(x^4) at (1,0)
Can someone please guide me through this problem?

amistre64 · Answer

product rule, and chain rule are used for the derivative

anonymous · Answer

how so, I know that the product rule is used to multiply both, but how is the chain rule used?

amistre64 · Answer

the chain rule is used since there are function inside of functions

amistre64 · Answer

$$D_x[f(g(x))]=f'(g(x))*g'(x)$$

anonymous · Answer

ok so then would the chain rule be 3 (-sinx) (3)?

anonymous · Answer

(3x)*

amistre64 · Answer

close, 

cos(3x)  ->  -sin(3x) * (3x)'  ->  -3sin(3x)

anonymous · Answer

ahh and so overall it would be (cos 3x)(4x^3/x^4) + (lnx^4)(-3sin3x)?

amistre64 · Answer

that looks good to me.  the left part can be simplified to (cos 3x)/x  but thats inconsequential

amistre64 · Answer

since ln1 = 0; the right side is useless to us; giving us:  cos(3) as the slope of the line at the given point

anonymous · Answer

ahh so I just have to simplify, plug in 1 for x, and then put it into pint-slope form, ok thank you so much, you have no idea how much this has helped me today. Especially, since I have a test on this tommorrow

amistre64 · Answer

good luck ;)

amistre64 · Answer

and i forgot a 4 in the type up :/

sooo, 4 cos(3)

anonymous · Answer

so that would be the slope? And thanks I'll be needing it! :)

amistre64 · Answer

yes, just remember that derivatives are slopes

anonymous · Answer

ok will do thanks